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<?xml version="1.0" standalone="yes"?> <Paper uid="C82-1003"> <Title>KNOWLEDGE REPRESENTATION METHOD BASED ON pREDICATE CALCULUS IN AN INTELLIGENT CAI SYSTEM</Title> <Section position="3" start_page="13" end_page="13" type="metho"> <SectionTitle> 14 B. BEGIER PREDICATE CALCULUS APPLIED TO REPRESENT EN01~EDGE ABOUT THE PRO- GRAMMING LANGUAGE </SectionTitle> <Paragraph position="0"> The followin~ criteria may be referred to knowledge representation methods for ICAI systems \[.1,7\]: - ability to express a large set of concepts of the domain bein~ * taught p - facility of codln~ these concepts and relations amon~ them~ - easy way to t~ansform the formal notation into the natur~l language formp - effiolenoy of information retrieval duri~ the process of answerin~ user's query and proving the correctness of his answer~ - ability of automated deduction application in the question answerln~ process.</Paragraph> <Paragraph position="1"> Let us conside~ a subset of an ALGOL-llke progx~ammin~ lan~uage~containin~ simple arithmetic and logical expressions~ instruction of substitution and conditional and o~ instructions. We assume that each instruction has been written in a separate line of program. The predicate calculus language developed to represent knowledge about the progI~ammlng lan6~age contains: - names of sets~ called sorts of objects~ representing elements of syntax and semantics of a pro~Tammln~ lan~uagep - funotions~ transformln~ obJeotsp - predioates~ representin~ relations between objects. Some sortsj functions and predicates are introduced to represent syntax of the prog~ammlng lan~uage.0thers represent its semantics. N o t a t i o n. The ordinary predicate calculus notation has been used. Some modifications improve the readability of statements: - unquantified vat-fables are gener~lly qua~t~.~le~. ~ - two-places predicates are written in an infix manner~ - binary arithmetic functions are written in an infix ma~n~rp - parenthesis are used in the ordinary meanin~ - clauses are separated by dots.</Paragraph> <Paragraph position="2"> P r o g r a m s y n t a x. The program syntax has been described by a set of clauses w~itten in the predicate calculus language. Sorts of objects /examples/~ identifier~ number~ expresslon~ arlthmetio expression~ logical expression~ label~ instruction~ program line.</Paragraph> <Paragraph position="3"> Functions transform some expressions_into te~ms~ by example: dod_~ wax wa-w-wa where: ~a - arithmetic expression~ it: wa~wa---wl wl - logical expression~ pod_~ id x wy~ in id - identifier~ sko; et~ in wy - expression~ et - label~ ifl: wl x wi ----in in - instruction~ ~ri - program line.</Paragraph> <Paragraph position="4"> First of these functions constructs an expression~ which represents an operation of addition~ the second one gives as a result an expression representing the &quot;less than&quot; relation and the others constr~et appropriately the substitution instr~ction~ the ~ instruction and the conditional instruction.</Paragraph> <Paragraph position="5"> Some p~edlcates have been introduced to represent the syntax relations between syntax obJeots~ like followin~:</Paragraph> <Paragraph position="7"> First of these predicates indicates the location of an instruction</Paragraph> </Section> <Section position="4" start_page="13" end_page="13" type="metho"> <SectionTitle> KNOWLEDGE REPRESENTATION METHOD IN AN ICAI SYSTEM 15 </SectionTitle> <Paragraph position="0"> in a given pro~z~m llne r the second one assi~s a label at the be6~i~ning of a progmam line and the third one determines the direct succession of two progx~um lines in a sequence.</Paragraph> <Paragraph position="1"> Example,. The syntax of a pro~am containing three followin~ substi-</Paragraph> <Paragraph position="3"> guage is represented by a set of axioms ~rit%en in the predicate calculus l~age. New sorts of obJects~ functions and predicates are to be intrOduced.</Paragraph> <Paragraph position="4"> Sorts of ob~eetsz value~ state Value set contains values of ax-lthmeti0 and logical expresslons~ar~ys~ subroutines or procedures etCo The particular kind of value is a location in a pro~nam~ represented by a progr~um llne. A state * is assiEned to a progx~un line and it is determined also by the ca. luation of variables.</Paragraph> <Paragraph position="5"> Functions evaluate expressions * owar: wy X st --'~T where: wy - expression~ and sequence of states: st ~ state~ haS: st ~st w~ - ~lueo Predicates assign states to iooatior~ in a program: stwe c-st ~wi stzm ~st xwi Pirs$,c~&quot; them associates a state to a program line before an execution of an instruction from this lineo The second one indicateo 9 that the control p~ssee ~o an instruction ~ri%ten in a ~iven program line in a ~iven state.</Paragraph> <Paragraph position="6"> Sorts o~ ob~eets~ functions and predicates are the b~is o~ a ~r~mmar of ~he predicate cal~ulus lang~a~e~ which expreu~ions are ~sed before an execution o. ~. %~.~is instrue~ion~and a faot~that the program line w2 directly follo~s&quot;wlo The ooncl~slo~ says~ %.h~ a s~;ate next of s as %he state before an exertion of an instruction written in ~2 and 8 value of the variable J in the state next of s is a value of an exp~-ession X in the ~tate s ~nd ~ ~lue of any variable Y ~ J doesn't ehan~e during %he %rar~fer from the state s to the next of ~o A l~r~e subse~ oPS FORTRAN has been de$~rlbed in this manner \[~\] deg it tul~n8 out ~hat form%11as of p~edicate calculus a~e easy to tr~ns-</Paragraph> </Section> <Section position="5" start_page="13" end_page="13" type="metho"> <SectionTitle> 16 'B. BEGIER </SectionTitle> <Paragraph position="0"> form into na%~ral la~ e~osPS~. Axles are divided into simple sentences. Translati~ rales are applied to simple sentences.</Paragraph> <Paragraph position="1"> Each object has and me in natur~l l~ge. Also an appropriate ~%ural language expression is selected for eaohfunotion.Eaah predicate cOrTesponds with a verb phrase in natural language. The proper translation rules for functions and objects sure applied with reference te arguments of a predloate.~lation ~les for Polish language have been reported in \[I\] as well as their a~plication to all axioms descrlbin~ FORTRAN.</Paragraph> <Paragraph position="2"> DIRECTED GRAPH AS A D~rHOD OF INSTRUCTIONAL STRUCTURE REPRESENTATION An assumption is done that all knowledge to be tau6ht can be divided to instructional units.Thus the first step to oonstr~ot an instructional structure representation is to select such units /concepts/. Each of them has a name and at least one sentence can be told about it /unreal concepts are not allowed/.Some introductory concepts are assumed to be known to the student.</Paragraph> <Paragraph position="3"> The next step is to specify all relations between concepts.These re- null Each relation corresponds with a graph, which nodes represent concepts from an introduced set of concepts.The composition of all obtained 6~aphs rerults in a final graph~whlch represents an instructional structure of the subject matter.Because of the different interpretation of the particular arches of this ~Taph /which are described by various relationships/ the &quot;superior-inferior&quot; relation is introduced as the universal one which represents every relation between concepts.Thus the dlrested ~phhas been obtained,~ith arrows directed to the superior concepts.</Paragraph> <Paragraph position="4"> A set of axioms is associated with each node of the concept graph.</Paragraph> <Paragraph position="5"> Also some other information may be associated with it.</Paragraph> </Section> <Section position="6" start_page="13" end_page="13" type="metho"> <SectionTitle> ANSWERING STUDENT'S QUESTIONS </SectionTitle> <Paragraph position="0"> The followin~ problems have to be solved| - choice and specifyin~ of classes of user's queries,which can be answered by the ICAI system~ - reco@~nition of a main subject of the query, translation of the query from natural language to the predicate calculus language formula, - application of the automated theorem-provin~ techniques to retrieve an answer, - generatin~ of an answer in natura~ language form. Three classes of queries have been qonsidered: /I/ Decision queries of gener~l\form in natural language <interrogative particle> < sentence> 7 where: <interrogative partiole>/existin~ in Polish/ determines that a question belongs to this class 9 ~entenoe> - indicative sentence, KNOWLEDGE REPRESENTATION lVlETHOD IN AN ICAI SYSTEM 17 which require an answer in the form .Y~s, or &quot;No&quot;. /2/ Objective quez-lee of ~enez~al foz~ in natuz~l lansuage Whloh ~senex~l name> <predloate~ ? whloh require to retrieve an object posessi~E some glven features as an answer.</Paragraph> <Paragraph position="1"> The quez 7 of this class may be tz~nsformed into the forml Which X satisfies a oo~Litlon: W (X) ~ A ~) ? whePez Y - an object to be foundp W ~ - distinctive predicate of a set~ which is specified by ~genez~L1 name> in the query, A(X) -. formula obtained iron the tz~nslated queryp which describes some prope~ties of X.</Paragraph> <Paragraph position="2"> /3/ Problem quex-ies of general form in natural language /a/ ~ <clause> ? whloh can be t~sfo .treed into the form: Which Z eatlsfies: Z~ ~clause> ? /b/ What is implied by ~elause> ? which can be tz~nsfo~med into the formz Which Z satisfies = ~olause~Z ? In the above problem queries: Z - the clause to be found t ~olause~ - clause obta/ned from the tx~nslate~ query. P~oblem queries ~equlre an answer in the foz~ of a sentenCOo null An analysis Of a userts quex-y should flx the main subject of it in the terms of a subset of conoepts~represented on the concept g~ph. A dofiD/tlon of acceptable language of user's queries involves the form of translation rules from n~tux~tl lan~u.age into the predloate calculus formula. It is worth notloln~ that: - querles in the natural lar~ua~e PSorm have the threefold nature~ it means they can be counted into the three mentioned above olasses~ - queries fragments in the form of indicative clauses are built from expressed in natural language p~edloates~introduee~ in the present ed f or~alization~ - in respect of quantity of expressions the language of user's queries is comparable with the laa1&~e obtained in the process of translation of p~ed/oate calculus axioms into natural lan~ua~e~ - lan@uaGe o~ user's queries and the predloate calculus l~a~e have a common base of basle concepts because the sorts of ob~eots~ functions and pFedicates introduced in the predloate calculus language correspond with some specified natural language expressions. It has been submitted that questlon-answering problem m~y be solved with an application o~ auto~te~ theorem-pyo~,in~ teohnlques~ namely on the base o\[ the resolution principle \[8\] ,This method~ based only on the s~rnteac of olauses~ doesn't require to control the proo~ procedure by the user,The resolution prlnelple requires to convert all formulas into the Skolem conjunctive form. Thus each formula becomes a set of olauses~ each of them bei~E a dlsjunotion of literals, The question-ans~ering procedure for deolslon queries tries to pro~e that a negation of formula ~ ob%alned after translation of the quer~ is false, IPS it's so~ an answer is &quot;Yes&quot;. If a proo~ procedure applied to the formula in its a~firmati~/e ~or~ provides a sucoess~ an answer is &quot;NO&quot;.Some questions may be unsolvable in the lack of kno~le~e.</Paragraph> <Paragraph position="3"> The proof procedure PSor objective queries examines a PSormula ~~x (w (x} ~ A (x)) 18 B. BEGIER which is supposed to be false.The proof prroedure tries to retrieve a counter-example, if i% exists, which will be substituted in the place of X o I% has been assumed that problem queries are in the implication form after the translation proeess. Question-answerin~ procedure for this class of queries has been reduced to such onetwhich tries to retrieve an answer from one axiom. For the first subclass of problem queries a search is made for an axiom in the implication form, ~hich conclusion embodies the conclusion of the formula obtained from the tx~ansformed query. The premises of this axiom are an answerdeg The proof procedure applied to answer a question of the second subclass tries to find an axiom~ which premises are implied by premises of the formula obtained f~om the transformed query. The conclusion of this axiom is an answer.</Paragraph> <Paragraph position="4"> The proper way to reduce a number of clauses taking a part in the resolution process is to constr,~ct an initial active set of clauses as a set containing only clauses of axioms concerning concepts reco~LIzed in the query and clauses of formula obtained from the query. The translation rules,applied to transform axioms from the predicate calculus language into the natural language expressions t can be used also to translate the retrieved answer to the natural languagedeg</Paragraph> </Section> class="xml-element"></Paper>